Solution of the truncated hyperbolic moment problem

Abstract

Let Q(x,y)=0 be an hyperbola in the plane. Given real numbers $\beta \equiv {\beta }^{2n)}=\{{\beta }_{ij}{\}}_{i,j\ge 0,i+j\le 2n}$, with ${\beta }_{00}>0$, the truncated Q-hyperbolic moment problem for βentails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x,y)=0, such that ${\beta }_{ij}=\int y^i{x}^{j}d\mu (0\le i+j\le 2n)$. We prove that βadmits a Q-representing measure μ(as above) if and only if the associated moment matrix $\mathcal{M}(n)(\beta )$ is positive semidefinite, recursively generated, has a column relation Q(X,Y)=0, and the algebraic variety $\mathcal{V}(\beta )$ associated to βsatisfies $card\mathcal{V}(β)\geq\rank\mathcal{M}(n)(β)$. In this case, $rank\mathcal{M}(n)\le 2n+1$; if $rank\mathcal{M}(n)\le 2n$, then βadmits a $rank\mathcal{M}(n)$-atomic (minimal) Q-representing measure; if $rank\mathcal{M}(n)=2n+1$, then βadmits a Q-representing measure μsatisfying $2n+1\leqcard suppμ\leq2n+2$.